It is already well known that tolerances of important influence factors must be taken into account for planning a technical system or a technical product. Conventionally high security margins are provided to take no risk respectively to take risks as low as possible for the planning of a system. This may lead to high fabrication and/or operation costs. For some special applications software packages do exist:
COMREL
COMREL is based is based on FORM/SORM and exists in two variants (FORM/SORM are first and second order reliability methods). COMREL is for reliability analysis of components. COMREL consists of two parts: COMREL-TI for time invariant and COMREL-TV for time variant reliability analysis. Base for both program parts is the method of first order (FORM) or second order (SORM). COMREL-TI can be supplied separately. COMREL-TV bases on COMREL-TI. COMREL uses two alternative, efficient and robust algorithms to find the so called beta-point (point for the local highest constraint or failure probability). The better-point is the base for the FORM/SORM method for a probability integration. Other options for probability integration are mean value first order (MVFO), Monte Carlo simulation, adaptive simulation, spheric simulation and several importance sampling schemata. 44 different probability distributions (univariate stochastic models) are useable. Arbitrary dependency structures can be generated with the aid of the Rosenblatt-transformation, the equivalent correlation coefficients according to Nataf, Der Kiuerghian or of the Hermite-models. Next to the reliability index importance values for all relevant input values are calculated: Global influence of the basis variables to the reliability index, sensitivities and elasticities, for the distribution parameters, the mean values and the standard deviations of the basis variables, sensitivities and elasticities for deterministic parameters in the constraint or failure function. Out of the sensitivity analysis partial security factors are deviated. Parameter studies can be performed for arbitrary values, e.g. for a distribution parameter, a correlation coefficient or a deterministic parameter. Basing on a parameter study charts of the reliability index, of the failure or of the survival probability, of the influences of basic variables or of deterministic parameters and of the expectancy value of a cost function can be generated. All results are available as a structured text file and as a file for the generation of plots. Charts generated in COMREL and formatted with the extensive plot options can be exported with the usual Windows equipment (Clipboard, Metafile, Bitmap). If it is necessary, a detailed outprint of provisional results for a failure search can be generated.
NESSUS
NESSUS (Numerical Evaluation of Stochastic Structures Under Stress) is an integrated finite element program with probabilistic load. Probabilistic load. Probabilistic sensitivities relating to μ and σ: FORM/SORM/FPI (fast probability integration), . . . connection to ANSYS, ABAQUS, DYNA3D. NESSUS is a modular computer software system for performing probabilistic analysis of structural/mechanical components and systems. NESSUS combines state of the art probabilistic algorithms with general-purpose numerical analysis methods to compute the probabilistic response and reliability of engineered systems. Uncertainty in loading, material properties, geometry, boundary conditions and initial conditions can be simulated. Many deterministic modeling tools can be used such as finite element, boundary element, hydrocodes, and user-defined Fortran subroutines.
DARWIN
DARWIN (Design Assessment of Reliability With Inspection)
This software integrates finite element stress analysis results, fracture-mechanics based life assessment for low-cycle fatigue, material anomaly data, probability of anomaly detection and inspection schedules to determine the probability of fracture of a rotor disc as a function of applied operating cycles. The program also indicates the regions of the disk most likely to fail, and the risk reduction associated with single and multiple inspections. This software will be enhanced to handle anomalies in cast/wrought and powder nickel disks and manufacturing and maintenance-induced surface defects in all disk materials in the near future.
The programs NESSUS, DARWIN and COMREL have a certain distribution in industry. All those programs merely concern mechanical reliability analyze. Finite element packages are integrated, in which stochastic is directly integrated. Thus the stochastic distribution of the load can directly be converted into the distributions of the displacements and a component part can be converted into risk zones by this. For elected, external finite element programs there exist interfaces at NESSUS and COMREL. DARWIN and COMREL moreover merely offer an instationary analysis. The process variables can be stochastic processes on a limited scale. Stochastic optimization, is not integrated within NESSUS, DARWIN and COMREL.
Nonlinear optimization algorithms cope with the problem:
                                          min                                          x                ->                            ∈                              ℛ                n                                              ⁢                      f            ⁡                          (                              x                ->                            )                                      ,                              g            ⁡                          (                              x                ->                            )                                ⁢                      >            _                    ⁢          0                ,                            (        1        )            with g({right arrow over (x)}) is a constraint, especially a failure, of an arbitrary value, the constraint or failure being caused by deterministic input parameters.
When {right arrow over (x)} are no longer deterministic variables but stochastic random variables (e.g. normal distributed random variable {right arrow over (x)} ∈ N({right arrow over (μ)}, Σ)) the deterministic optimization problem (1) passes into the following probabilistic optimization problem:
                                          min                                          x                ->                            ∈                              N                ⁡                                  (                                                            μ                      ->                                        ,                    Σ                                    )                                                              ⁢                      E            ⁡                          (                              f                ⁡                                  (                                      x                    ->                                    )                                            )                                      ,                              P            ⁡                          (                                                g                  ⁡                                      (                                          x                      ->                                        )                                                  ⁢                                  <                  _                                ⁢                0                            )                                ⁢                      <            _                    ⁢          tol                                    (        2        )            I.e. the expectation value respectively mean value of the target size, E(f({right arrow over (x)})), is minimized and the constraints may be violated up to a prescribed tol. The mean values {right arrow over (μ)} of the input parameters are the design parameters. FIG. 1 illustrates the optimization problems (1) and (2). The deterministic optimization problem passes into the probabilistic optimization problem. When design variables respectively input parameters are afflicted with uncertainties the optional dimensioning of a technical system y=f({right arrow over (x)}) leads to a new and different operating point. In other words minimizing the function f({right arrow over (x)}) leads to results that are different from those obtained by minimizing the mean value of the function f({right arrow over (x)}). This effect may be observed in FIG. 2. FIG. 2 shows an optimal dimensioning of the system y=f({right arrow over (x)}), whereby an deterministic optimization is shown left, a representation of the constraint or the failure in case of a deterministic optimization is shown in the middle and a probabilistic optimization of the system is shown on the right.
A popular method for the computation of the response of a stochastic system is the Monte-Carlo method. The computation of the mean value and the variance of the system y=f({right arrow over (x)}) is presented in the following Table:
Monte Carlo method:
START: Determine a set {right arrow over (x)}1, . . . , {right arrow over (x)}m, which represents the distribution of the input parameter.
                                                        ITERATE              ⁢                              :                            ⁢                                                          ⁢              j                        =            1                    ,          2          ,          3          ,          ...                                          ,          m                ⁢                                  ⁢                              y            j                    =                      f            ⁡                          (                                                x                  ->                                j                            )                                      ⁢                                  ⁢                  END          ⁢                                          ⁢          j                ⁢                                  ⁢                              EVALUATE            ⁢                          :                        ⁢                                                  ⁢                          E              ⁡                              (                Y                )                                              =                                    1              m                        ⁢                                          ∑                                  1                  ⁢                                      <                    _                                    ⁢                  j                  ⁢                                      <                    _                                    ⁢                  m                                            ⁢                              y                j                                                    ⁢                                  ⁢                              EVALUATE            ⁢                          :                        ⁢                                                  ⁢                          V              ⁡                              (                Y                )                                              =                                    1                              m                -                1                                      ⁢                                          ∑                                  1                  ⁢                                      <                    _                                    ⁢                  j                  ⁢                                      <                    _                                    ⁢                  m                                            ⁢                                                (                                                            y                      j                                        -                                          E                      ⁡                                              (                        Y                        )                                                                              )                                2                                                                                    
In order to assure a correct computation of the characteristic sizes E(Y) and V(Y), the size m of the ensemble must be very large. Hence, embedding the Monte-Carlo method into a framework of optimization is difficult in practical cases: To handle computational fluid dynamics or large Finite Element problems in reasonable time neither a super computer nor a large cluster of workstations would suffice.